Let Sn(x)= nx/(nx+1)
Compute the pointwise limit of Sn on [0,1].
Is the convergence uniform?
$\displaystyle S_n \to S$ point-wise on [0,1] where $\displaystyle S = \left \{ \begin{array}{lr} 0 & \mbox{ if } x = 0 \\ & \\ 1 & \mbox{ if } x \in (0,1] \end{array} \right.$
all we did was to take the limit as $\displaystyle n \to \infty$
Is it uniform? there are several ways to attack this.
what do we know about the limit of a uniformly continuous function?
we could also use the $\displaystyle \epsilon - \delta$ definition for uniform convergence
we could also check to see whether or not $\displaystyle \lim_{n \to \infty} [ \sup \{ |S(x) - S_n(x)|~:~x \in [0,1] \}] = 0$ in which case it is uniformly convergent if it is 0